Correlational Resource Theory of Catalytic Quantum Randomness under Conservation Law

TL;DR

This paper develops a resource theory for catalytic quantum randomness, characterizing the limits of entropy extraction via catalysts under conservation laws, and reveals how degeneracy and state properties influence catalytic entropy.

Contribution

It introduces a complete characterization of bipartite unitaries for catalytic randomness, defines catalytic entropies, and explores their behavior under conservation laws, highlighting new limitations and potentials.

Findings

Catalytic channels are factorizable, with some unital channels being non-catalytic.

Degeneracy in quantum states can enhance catalytic entropy beyond standard entropy.

Maximal catalytic entropy can be achieved by non-maximally mixed states under conservation laws.

Abstract

Catalysts are substances that assist transformation of other resourceful objects without being consumed in the process. However, the fact that their `catalytic power' is limited and can be depleted is often overlooked, especially in the recently developing theories on catalysis of quantum randomness utilizing building correlation with catalyst. In this work, we establish a resource theory of one-shot catalytic randomness in which uncorrelatedness is consumed in catalysis of randomness. We do so by completely characterizing bipartite unitary operators that can be used to implement catalysis of randomness using partial transpose. By doing so, we find that every catalytic channel is factorizable, and therefore there exists a unital channel that is not catalytic. We define a family of catalytic entropies that quantifies catalytically extractable entropy within a quantum state and show how much degeneracy of quantum state can boost the catalytic entropy beyond its ordinary entropy. Based on this, we demonstrate that a randomness source can be actually exhausted after a certain amount of randomness is extracted. We apply this theory to systems under conservation law that forbids superposition of certain quantum states and find that non-maximally mixed states can yield the maximal catalytic entropy. We discuss implications of this theory to various topics including catalytic randomness absorption, the no-secret theorem and the possibility of multi-party infinite catalysis.